Objective
Build a linear regression model to predict a car’s CO2 Emissions (grams / Km). The model will be built by using the “Fuel consumption ratings” dataset from the Government of Canada.
Project Overview
We made a descriptive analysis to understand the data and decide which variables could be dropped to proceed with the modelling section.
We split the data into 80% for training and 20% for testing.
We applied the variable screening methods to the train data, and all methods suggested a full model which though high in r-squared value it had a pattern in the residual plot. The pattern was eliminated with an interaction and named the model as “Model 8”.
We ran some models based on the descriptive analysis, and chose to use a simple model which still had high accuracy, and named the model as “Model 9”.
We fitted the test data into models 8 and 9 and compared the prediction for the two models.
Descriptive Analysis
Numerical variables
- YEAR
- ENGINE
- CYLINDERS
- CONSUMPTION
- EMISSIONS (Response Variable)
Correlation Matrix
Insights
EMISSIONS has three significant correlations with the other three numerical variables:
Emissions vs Fuel Consumption: 0.935
Emissions vs Engine: 0.799
Emissions vs Cylinders: 0.766
CYLINDERS has a high correlation with ENGINE: 0.910
Categorical Variables
Brand: 55 levels.
Model: 4185 levels.
Class: 17 levels.
Transmission: 5 levels.
Fuel: 5 levels.
Gears: 9 levels.
We are not going to consider Model in the Analysis, as this variables has a large number of categories, and it not provide useful information to predict emissions.
FUEL
Insights:
- The median CO2 emission for premium, regular and diesel are very close from each other, and they are surpassed by Ethanol and Natural Gas.
Insights:
- The majority of cars in the data set uses gasoline, which can be premium or regular in similar proportions. Ethanol grew in 2007 but the number was reduced by 2015.
Insights:
The majority of cars uses gasoline and in consumption they are concentrated in the same line, so they could be considered as one type of gasoline. The lines are separated from each other, which makes them the perfect candidate for interaction, CONSUMPTION*FUEL
The bigger the engine size, the gasoline premium is more required.
From 10 cylinders and above the only fuel used is premium.
TRANSMISSION
Insights:
- Continuous variable has the lowest co2 emission. This type of automatic transmission uses pulleys and a steel band instead of traditional fixed gears. The CVT can change the gear ratio forever to maintain the engine running at peak efficiency. On the whole, the more gears offered in a typical automatic transmission, the better engine power is optimized.
Insights:
- Automatic with selected Shift transmission grew considerably since 2005, and now is the main offer in the market. This kind of transmission allows to choose between fully automatic shifting or semi-automatic, clutch-less shifting.
Insights:
In consumption vs emissions, all the transmission lines are close to each other, which means that all transmission types are dispersed.
In engine size vs emissions, all the transmission lines are close to each other.
In cylinder vs emission, all the transmission lines are close to each other but continuous variable shift, which is also the most efficient for emission. We might see some interaction between transmission*cylinders, all the the effect would be solely in continuous variable shift.
CLASS
- There are 17 levels in CLASS, so with the purpose of analyze a way to consider this variable in the model, we are going to consider a way to unify categories and reduce the number of levels.
Insights
- In the case of unifying CLASS by the name, for example: Pickup Truck, Station Wagon, SUV, VAN, it would not be appropriate, since among these exist differences in the median emissions.
Insights
- There is no clear pattern if trying to unified classes by cylinders. There are many classes per each discrete number in the variable.
Insights
- There is no clear pattern if trying to unified classes by engine size.
Conclusion
- We are dropping the variable CLASS as a possible candidate to predictor in the model.
GEARS
Insights
Filled the NA values in Gears with 1 after analyzing that all the empty values corresponded to the Continuous Variable TRANSMISSION.
For the purpose of consider GEARS in the model, it will be considered as numerical instead of categorical, and limit the number of coefficients (dummy variables).
Insights
- GEARS is not greatly correlated with any of the other numerical variables.
BRAND
## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
Insights
The Brand with the highest average emission is Bugatti, and in general the brands that produce the high-end cars.
Never the less, for its high amount of levels, we are dropping the variable for further analysis.
Data preparation
Training and Testing Data
set.seed(123)
test_index = sample(seq_len(nrow(co2_filter)),size=0.2*nrow(co2_filter))
co2_test = co2_filter[test_index, ]
co2_train = co2_filter[-test_index,]
write.csv(co2_train, "train.csv")
write.csv(co2_test, "test.csv")attach(co2_train)Software model
Stepwise Regression
Both: Step
step(lm(EMISSIONS~.,data = co2_train),direction = "both")## Start: AIC=64849.83
## EMISSIONS ~ YEAR + ENGINE + CYLINDERS + TRANSMISSION + GEARS +
## FUEL + CONSUMPTION
##
## Df Sum of Sq RSS AIC
## - CYLINDERS 1 1 466522 64848
## - ENGINE 1 31 466552 64849
## <none> 466521 64850
## - TRANSMISSION 4 2476 468997 64952
## - YEAR 1 4811 471332 65062
## - GEARS 1 7135 473656 65164
## - FUEL 4 8533436 8999957 126581
## - CONSUMPTION 1 24657719 25124241 148002
##
## Step: AIC=64847.86
## EMISSIONS ~ YEAR + ENGINE + TRANSMISSION + GEARS + FUEL + CONSUMPTION
##
## Df Sum of Sq RSS AIC
## <none> 466522 64848
## - ENGINE 1 59 466581 64849
## + CYLINDERS 1 1 466521 64850
## - TRANSMISSION 4 2505 469027 64952
## - YEAR 1 4823 471345 65060
## - GEARS 1 7172 473694 65164
## - FUEL 4 8591934 9058455 126714
## - CONSUMPTION 1 25194831 25661353 148441##
## Call:
## lm(formula = EMISSIONS ~ YEAR + ENGINE + TRANSMISSION + GEARS +
## FUEL + CONSUMPTION, data = co2_train)
##
## Coefficients:
## (Intercept) YEAR
## -170.72666 0.10218
## ENGINE TRANSMISSIONAutomatic
## -0.07035 -0.63570
## TRANSMISSIONAutomaticSShift TRANSMISSIONContinuosVariable
## -0.69797 1.55180
## TRANSMISSIONManual GEARS
## -0.98620 0.66221
## FUELEtanol FUELNaturalGas
## -151.11978 -103.28606
## FUELPremium FUELRegular
## -35.14984 -34.62523
## CONSUMPTION
## 22.88428Both: MASS
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectstepAIC(lm(EMISSIONS~.,data=co2_train),direction="both")## Start: AIC=64849.83
## EMISSIONS ~ YEAR + ENGINE + CYLINDERS + TRANSMISSION + GEARS +
## FUEL + CONSUMPTION
##
## Df Sum of Sq RSS AIC
## - CYLINDERS 1 1 466522 64848
## - ENGINE 1 31 466552 64849
## <none> 466521 64850
## - TRANSMISSION 4 2476 468997 64952
## - YEAR 1 4811 471332 65062
## - GEARS 1 7135 473656 65164
## - FUEL 4 8533436 8999957 126581
## - CONSUMPTION 1 24657719 25124241 148002
##
## Step: AIC=64847.86
## EMISSIONS ~ YEAR + ENGINE + TRANSMISSION + GEARS + FUEL + CONSUMPTION
##
## Df Sum of Sq RSS AIC
## <none> 466522 64848
## - ENGINE 1 59 466581 64849
## + CYLINDERS 1 1 466521 64850
## - TRANSMISSION 4 2505 469027 64952
## - YEAR 1 4823 471345 65060
## - GEARS 1 7172 473694 65164
## - FUEL 4 8591934 9058455 126714
## - CONSUMPTION 1 25194831 25661353 148441##
## Call:
## lm(formula = EMISSIONS ~ YEAR + ENGINE + TRANSMISSION + GEARS +
## FUEL + CONSUMPTION, data = co2_train)
##
## Coefficients:
## (Intercept) YEAR
## -170.72666 0.10218
## ENGINE TRANSMISSIONAutomatic
## -0.07035 -0.63570
## TRANSMISSIONAutomaticSShift TRANSMISSIONContinuosVariable
## -0.69797 1.55180
## TRANSMISSIONManual GEARS
## -0.98620 0.66221
## FUELEtanol FUELNaturalGas
## -151.11978 -103.28606
## FUELPremium FUELRegular
## -35.14984 -34.62523
## CONSUMPTION
## 22.88428Forward: List
mfl = lm(EMISSIONS~1,data=co2_train)
forwardAIC = step(mfl,scope=list(lower=~1,
upper= ~ CONSUMPTION + CYLINDERS + ENGINE + FUEL + GEARS + TRANSMISSION + YEAR),
direction="forward", data=co2_train)## Start: AIC=172873.3
## EMISSIONS ~ 1
##
## Df Sum of Sq RSS AIC
## + CONSUMPTION 1 72525464 10346035 129472
## + ENGINE 1 52862466 30009032 151686
## + CYLINDERS 1 48825735 34045763 154319
## + TRANSMISSION 4 11684751 71186747 169711
## + YEAR 1 3409758 79461741 171999
## + FUEL 4 1238568 81632930 172567
## + GEARS 1 930019 81941479 172640
## <none> 82871498 172873
##
## Step: AIC=129472.1
## EMISSIONS ~ CONSUMPTION
##
## Df Sum of Sq RSS AIC
## + FUEL 4 9828471 517563 67000
## + CYLINDERS 1 766300 9579734 127869
## + ENGINE 1 698994 9647040 128015
## + GEARS 1 86144 10259891 129300
## + TRANSMISSION 4 85830 10260205 129306
## + YEAR 1 70951 10275083 129331
## <none> 10346035 129472
##
## Step: AIC=66999.68
## EMISSIONS ~ CONSUMPTION + FUEL
##
## Df Sum of Sq RSS AIC
## + YEAR 1 41911 475653 65240
## + GEARS 1 31612 485952 65687
## + TRANSMISSION 4 11879 505685 66523
## + ENGINE 1 850 516714 66967
## + CYLINDERS 1 846 516718 66968
## <none> 517563 67000
##
## Step: AIC=65240.18
## EMISSIONS ~ CONSUMPTION + FUEL + YEAR
##
## Df Sum of Sq RSS AIC
## + GEARS 1 6597.5 469055 64951
## + TRANSMISSION 4 1886.5 473766 65165
## + ENGINE 1 149.7 475503 65236
## <none> 475653 65240
## + CYLINDERS 1 15.5 475637 65241
##
## Step: AIC=64950.81
## EMISSIONS ~ CONSUMPTION + FUEL + YEAR + GEARS
##
## Df Sum of Sq RSS AIC
## + TRANSMISSION 4 2473.71 466581 64849
## <none> 469055 64951
## + ENGINE 1 28.06 469027 64952
## + CYLINDERS 1 0.31 469055 64953
##
## Step: AIC=64848.51
## EMISSIONS ~ CONSUMPTION + FUEL + YEAR + GEARS + TRANSMISSION
##
## Df Sum of Sq RSS AIC
## + ENGINE 1 59.419 466522 64848
## <none> 466581 64849
## + CYLINDERS 1 29.058 466552 64849
##
## Step: AIC=64847.86
## EMISSIONS ~ CONSUMPTION + FUEL + YEAR + GEARS + TRANSMISSION +
## ENGINE
##
## Df Sum of Sq RSS AIC
## <none> 466522 64848
## + CYLINDERS 1 0.62756 466521 64850Forward: formula
mff=lm(EMISSIONS~CONSUMPTION + CYLINDERS + ENGINE + FUEL + GEARS + TRANSMISSION + YEAR)
forwardAIC1 <- step(mfl,scope=formula(mff),
direction="forward", data=co2_train)## Start: AIC=172873.3
## EMISSIONS ~ 1
##
## Df Sum of Sq RSS AIC
## + CONSUMPTION 1 72525464 10346035 129472
## + ENGINE 1 52862466 30009032 151686
## + CYLINDERS 1 48825735 34045763 154319
## + TRANSMISSION 4 11684751 71186747 169711
## + YEAR 1 3409758 79461741 171999
## + FUEL 4 1238568 81632930 172567
## + GEARS 1 930019 81941479 172640
## <none> 82871498 172873
##
## Step: AIC=129472.1
## EMISSIONS ~ CONSUMPTION
##
## Df Sum of Sq RSS AIC
## + FUEL 4 9828471 517563 67000
## + CYLINDERS 1 766300 9579734 127869
## + ENGINE 1 698994 9647040 128015
## + GEARS 1 86144 10259891 129300
## + TRANSMISSION 4 85830 10260205 129306
## + YEAR 1 70951 10275083 129331
## <none> 10346035 129472
##
## Step: AIC=66999.68
## EMISSIONS ~ CONSUMPTION + FUEL
##
## Df Sum of Sq RSS AIC
## + YEAR 1 41911 475653 65240
## + GEARS 1 31612 485952 65687
## + TRANSMISSION 4 11879 505685 66523
## + ENGINE 1 850 516714 66967
## + CYLINDERS 1 846 516718 66968
## <none> 517563 67000
##
## Step: AIC=65240.18
## EMISSIONS ~ CONSUMPTION + FUEL + YEAR
##
## Df Sum of Sq RSS AIC
## + GEARS 1 6597.5 469055 64951
## + TRANSMISSION 4 1886.5 473766 65165
## + ENGINE 1 149.7 475503 65236
## <none> 475653 65240
## + CYLINDERS 1 15.5 475637 65241
##
## Step: AIC=64950.81
## EMISSIONS ~ CONSUMPTION + FUEL + YEAR + GEARS
##
## Df Sum of Sq RSS AIC
## + TRANSMISSION 4 2473.71 466581 64849
## <none> 469055 64951
## + ENGINE 1 28.06 469027 64952
## + CYLINDERS 1 0.31 469055 64953
##
## Step: AIC=64848.51
## EMISSIONS ~ CONSUMPTION + FUEL + YEAR + GEARS + TRANSMISSION
##
## Df Sum of Sq RSS AIC
## + ENGINE 1 59.419 466522 64848
## <none> 466581 64849
## + CYLINDERS 1 29.058 466552 64849
##
## Step: AIC=64847.86
## EMISSIONS ~ CONSUMPTION + FUEL + YEAR + GEARS + TRANSMISSION +
## ENGINE
##
## Df Sum of Sq RSS AIC
## <none> 466522 64848
## + CYLINDERS 1 0.62756 466521 64850forwardAIC1$coefficients## (Intercept) CONSUMPTION
## -170.72665847 22.88427986
## FUELEtanol FUELNaturalGas
## -151.11978335 -103.28606140
## FUELPremium FUELRegular
## -35.14983959 -34.62523342
## YEAR GEARS
## 0.10218330 0.66220824
## TRANSMISSIONAutomatic TRANSMISSIONAutomaticSShift
## -0.63570067 -0.69797284
## TRANSMISSIONContinuosVariable TRANSMISSIONManual
## 1.55179737 -0.98620435
## ENGINE
## -0.07034717forwardAIC1$anova## Step Df Deviance Resid. Df Resid. Dev AIC
## 1 NA NA 20859 82871498.1 172873.26
## 2 + CONSUMPTION -1 7.252546e+07 20858 10346034.6 129472.11
## 3 + FUEL -4 9.828471e+06 20854 517563.3 66999.68
## 4 + YEAR -1 4.191075e+04 20853 475652.6 65240.18
## 5 + GEARS -1 6.597537e+03 20852 469055.0 64950.81
## 6 + TRANSMISSION -4 2.473711e+03 20848 466581.3 64848.51
## 7 + ENGINE -1 5.941885e+01 20847 466521.9 64847.86Backward: Step
step(lm(EMISSIONS~.,data = co2_train),direction = "backward")## Start: AIC=64849.83
## EMISSIONS ~ YEAR + ENGINE + CYLINDERS + TRANSMISSION + GEARS +
## FUEL + CONSUMPTION
##
## Df Sum of Sq RSS AIC
## - CYLINDERS 1 1 466522 64848
## - ENGINE 1 31 466552 64849
## <none> 466521 64850
## - TRANSMISSION 4 2476 468997 64952
## - YEAR 1 4811 471332 65062
## - GEARS 1 7135 473656 65164
## - FUEL 4 8533436 8999957 126581
## - CONSUMPTION 1 24657719 25124241 148002
##
## Step: AIC=64847.86
## EMISSIONS ~ YEAR + ENGINE + TRANSMISSION + GEARS + FUEL + CONSUMPTION
##
## Df Sum of Sq RSS AIC
## <none> 466522 64848
## - ENGINE 1 59 466581 64849
## - TRANSMISSION 4 2505 469027 64952
## - YEAR 1 4823 471345 65060
## - GEARS 1 7172 473694 65164
## - FUEL 4 8591934 9058455 126714
## - CONSUMPTION 1 25194831 25661353 148441##
## Call:
## lm(formula = EMISSIONS ~ YEAR + ENGINE + TRANSMISSION + GEARS +
## FUEL + CONSUMPTION, data = co2_train)
##
## Coefficients:
## (Intercept) YEAR
## -170.72666 0.10218
## ENGINE TRANSMISSIONAutomatic
## -0.07035 -0.63570
## TRANSMISSIONAutomaticSShift TRANSMISSIONContinuosVariable
## -0.69797 1.55180
## TRANSMISSIONManual GEARS
## -0.98620 0.66221
## FUELEtanol FUELNaturalGas
## -151.11978 -103.28606
## FUELPremium FUELRegular
## -35.14984 -34.62523
## CONSUMPTION
## 22.88428All Possible Regression Subset
Regsubset
library(leaps)
subset_model1 = regsubsets(EMISSIONS~CONSUMPTION + CYLINDERS + ENGINE + FUEL + GEARS + TRANSMISSION + YEAR, data = co2_train, nvmax = 15)
summary(subset_model1)## Subset selection object
## Call: regsubsets.formula(EMISSIONS ~ CONSUMPTION + CYLINDERS + ENGINE +
## FUEL + GEARS + TRANSMISSION + YEAR, data = co2_train, nvmax = 15)
## 13 Variables (and intercept)
## Forced in Forced out
## CONSUMPTION FALSE FALSE
## CYLINDERS FALSE FALSE
## ENGINE FALSE FALSE
## FUELEtanol FALSE FALSE
## FUELNaturalGas FALSE FALSE
## FUELPremium FALSE FALSE
## FUELRegular FALSE FALSE
## GEARS FALSE FALSE
## TRANSMISSIONAutomatic FALSE FALSE
## TRANSMISSIONAutomaticSShift FALSE FALSE
## TRANSMISSIONContinuosVariable FALSE FALSE
## TRANSMISSIONManual FALSE FALSE
## YEAR FALSE FALSE
## 1 subsets of each size up to 13
## Selection Algorithm: exhaustive
## CONSUMPTION CYLINDERS ENGINE FUELEtanol FUELNaturalGas FUELPremium
## 1 ( 1 ) "*" " " " " " " " " " "
## 2 ( 1 ) "*" " " " " "*" " " " "
## 3 ( 1 ) "*" " " " " "*" "*" " "
## 4 ( 1 ) "*" " " " " "*" " " "*"
## 5 ( 1 ) "*" " " " " "*" "*" "*"
## 6 ( 1 ) "*" " " " " "*" "*" "*"
## 7 ( 1 ) "*" " " " " "*" "*" "*"
## 8 ( 1 ) "*" " " " " "*" "*" "*"
## 9 ( 1 ) "*" " " " " "*" "*" "*"
## 10 ( 1 ) "*" " " "*" "*" "*" "*"
## 11 ( 1 ) "*" " " " " "*" "*" "*"
## 12 ( 1 ) "*" " " "*" "*" "*" "*"
## 13 ( 1 ) "*" "*" "*" "*" "*" "*"
## FUELRegular GEARS TRANSMISSIONAutomatic TRANSMISSIONAutomaticSShift
## 1 ( 1 ) " " " " " " " "
## 2 ( 1 ) " " " " " " " "
## 3 ( 1 ) " " " " " " " "
## 4 ( 1 ) "*" " " " " " "
## 5 ( 1 ) "*" " " " " " "
## 6 ( 1 ) "*" " " " " " "
## 7 ( 1 ) "*" "*" " " " "
## 8 ( 1 ) "*" "*" " " " "
## 9 ( 1 ) "*" "*" " " " "
## 10 ( 1 ) "*" "*" " " " "
## 11 ( 1 ) "*" "*" "*" "*"
## 12 ( 1 ) "*" "*" "*" "*"
## 13 ( 1 ) "*" "*" "*" "*"
## TRANSMISSIONContinuosVariable TRANSMISSIONManual YEAR
## 1 ( 1 ) " " " " " "
## 2 ( 1 ) " " " " " "
## 3 ( 1 ) " " " " " "
## 4 ( 1 ) " " " " " "
## 5 ( 1 ) " " " " " "
## 6 ( 1 ) " " " " "*"
## 7 ( 1 ) " " " " "*"
## 8 ( 1 ) "*" " " "*"
## 9 ( 1 ) "*" "*" "*"
## 10 ( 1 ) "*" "*" "*"
## 11 ( 1 ) "*" "*" "*"
## 12 ( 1 ) "*" "*" "*"
## 13 ( 1 ) "*" "*" "*"Results
SUM = summary(subset_model1)
names(SUM)## [1] "which" "rsq" "rss" "adjr2" "cp" "bic" "outmat" "obj"Rsq=SUM$rsq
CP=SUM$cp
AdRsq=SUM$adjr2
BIC=SUM$bic
RSS=SUM$rss
cbind(SUM$which,round(cbind(Rsq,AdRsq,CP,BIC,RSS),4))## (Intercept) CONSUMPTION CYLINDERS ENGINE FUELEtanol FUELNaturalGas
## 1 1 1 0 0 0 0
## 2 1 1 0 0 1 0
## 3 1 1 0 0 1 1
## 4 1 1 0 0 1 0
## 5 1 1 0 0 1 1
## 6 1 1 0 0 1 1
## 7 1 1 0 0 1 1
## 8 1 1 0 0 1 1
## 9 1 1 0 0 1 1
## 10 1 1 0 1 1 1
## 11 1 1 0 0 1 1
## 12 1 1 0 1 1 1
## 13 1 1 1 1 1 1
## FUELPremium FUELRegular GEARS TRANSMISSIONAutomatic
## 1 0 0 0 0
## 2 0 0 0 0
## 3 0 0 0 0
## 4 1 1 0 0
## 5 1 1 0 0
## 6 1 1 0 0
## 7 1 1 1 0
## 8 1 1 1 0
## 9 1 1 1 0
## 10 1 1 1 0
## 11 1 1 1 1
## 12 1 1 1 1
## 13 1 1 1 1
## TRANSMISSIONAutomaticSShift TRANSMISSIONContinuosVariable TRANSMISSIONManual
## 1 0 0 0
## 2 0 0 0
## 3 0 0 0
## 4 0 0 0
## 5 0 0 0
## 6 0 0 0
## 7 0 0 0
## 8 0 1 0
## 9 0 1 1
## 10 0 1 1
## 11 1 1 1
## 12 1 1 1
## 13 1 1 1
## YEAR Rsq AdRsq CP BIC RSS
## 1 0 0.8752 0.8751 441445.4143 -43383.26 10346034.6
## 2 0 0.9874 0.9874 25744.4960 -91239.69 1042847.0
## 3 0 0.9889 0.9889 20160.8681 -93893.20 917843.9
## 4 0 0.9906 0.9906 14023.4985 -97265.88 780448.4
## 5 0 0.9938 0.9938 2278.7587 -105823.91 517563.3
## 6 1 0.9943 0.9943 408.0221 -107575.47 475652.6
## 7 1 0.9943 0.9943 115.2182 -107856.88 469055.0
## 8 1 0.9944 0.9944 41.7123 -107922.22 467365.2
## 9 1 0.9944 0.9944 25.6971 -107930.28 466962.1
## 10 1 0.9944 0.9944 23.9186 -107924.11 466877.5
## 11 1 0.9944 0.9944 12.6831 -107927.40 466581.3
## 12 1 0.9944 0.9944 12.0280 -107920.12 466521.9
## 13 1 0.9944 0.9944 14.0000 -107910.20 466521.3n = length(co2_train$EMISSIONS) #number of observations
p = apply(SUM$which, 1, sum) #number of variables#Calculation of AIC
AIC = SUM$bic - log(n) * p + 2 * p
#number of independent variables in the models
I=p-1
MSE1=RSS/(n-I-1)SUM$which## (Intercept) CONSUMPTION CYLINDERS ENGINE FUELEtanol FUELNaturalGas
## 1 TRUE TRUE FALSE FALSE FALSE FALSE
## 2 TRUE TRUE FALSE FALSE TRUE FALSE
## 3 TRUE TRUE FALSE FALSE TRUE TRUE
## 4 TRUE TRUE FALSE FALSE TRUE FALSE
## 5 TRUE TRUE FALSE FALSE TRUE TRUE
## 6 TRUE TRUE FALSE FALSE TRUE TRUE
## 7 TRUE TRUE FALSE FALSE TRUE TRUE
## 8 TRUE TRUE FALSE FALSE TRUE TRUE
## 9 TRUE TRUE FALSE FALSE TRUE TRUE
## 10 TRUE TRUE FALSE TRUE TRUE TRUE
## 11 TRUE TRUE FALSE FALSE TRUE TRUE
## 12 TRUE TRUE FALSE TRUE TRUE TRUE
## 13 TRUE TRUE TRUE TRUE TRUE TRUE
## FUELPremium FUELRegular GEARS TRANSMISSIONAutomatic
## 1 FALSE FALSE FALSE FALSE
## 2 FALSE FALSE FALSE FALSE
## 3 FALSE FALSE FALSE FALSE
## 4 TRUE TRUE FALSE FALSE
## 5 TRUE TRUE FALSE FALSE
## 6 TRUE TRUE FALSE FALSE
## 7 TRUE TRUE TRUE FALSE
## 8 TRUE TRUE TRUE FALSE
## 9 TRUE TRUE TRUE FALSE
## 10 TRUE TRUE TRUE FALSE
## 11 TRUE TRUE TRUE TRUE
## 12 TRUE TRUE TRUE TRUE
## 13 TRUE TRUE TRUE TRUE
## TRANSMISSIONAutomaticSShift TRANSMISSIONContinuosVariable TRANSMISSIONManual
## 1 FALSE FALSE FALSE
## 2 FALSE FALSE FALSE
## 3 FALSE FALSE FALSE
## 4 FALSE FALSE FALSE
## 5 FALSE FALSE FALSE
## 6 FALSE FALSE FALSE
## 7 FALSE FALSE FALSE
## 8 FALSE TRUE FALSE
## 9 FALSE TRUE TRUE
## 10 FALSE TRUE TRUE
## 11 TRUE TRUE TRUE
## 12 TRUE TRUE TRUE
## 13 TRUE TRUE TRUE
## YEAR
## 1 FALSE
## 2 FALSE
## 3 FALSE
## 4 FALSE
## 5 FALSE
## 6 TRUE
## 7 TRUE
## 8 TRUE
## 9 TRUE
## 10 TRUE
## 11 TRUE
## 12 TRUE
## 13 TRUEresults_df = data.frame(
Model = SUM$which,
Rsq = round(Rsq, 4),
AdRsq = round(AdRsq, 4),
CP = round(CP, 4),
BIC = round(BIC, 4),
RSS = round(RSS, 4),
AIC = round(AIC, 4),
MSE1 = round(MSE1, 4)
)
results_df## Model..Intercept. Model.CONSUMPTION Model.CYLINDERS Model.ENGINE
## 1 TRUE TRUE FALSE FALSE
## 2 TRUE TRUE FALSE FALSE
## 3 TRUE TRUE FALSE FALSE
## 4 TRUE TRUE FALSE FALSE
## 5 TRUE TRUE FALSE FALSE
## 6 TRUE TRUE FALSE FALSE
## 7 TRUE TRUE FALSE FALSE
## 8 TRUE TRUE FALSE FALSE
## 9 TRUE TRUE FALSE FALSE
## 10 TRUE TRUE FALSE TRUE
## 11 TRUE TRUE FALSE FALSE
## 12 TRUE TRUE FALSE TRUE
## 13 TRUE TRUE TRUE TRUE
## Model.FUELEtanol Model.FUELNaturalGas Model.FUELPremium Model.FUELRegular
## 1 FALSE FALSE FALSE FALSE
## 2 TRUE FALSE FALSE FALSE
## 3 TRUE TRUE FALSE FALSE
## 4 TRUE FALSE TRUE TRUE
## 5 TRUE TRUE TRUE TRUE
## 6 TRUE TRUE TRUE TRUE
## 7 TRUE TRUE TRUE TRUE
## 8 TRUE TRUE TRUE TRUE
## 9 TRUE TRUE TRUE TRUE
## 10 TRUE TRUE TRUE TRUE
## 11 TRUE TRUE TRUE TRUE
## 12 TRUE TRUE TRUE TRUE
## 13 TRUE TRUE TRUE TRUE
## Model.GEARS Model.TRANSMISSIONAutomatic Model.TRANSMISSIONAutomaticSShift
## 1 FALSE FALSE FALSE
## 2 FALSE FALSE FALSE
## 3 FALSE FALSE FALSE
## 4 FALSE FALSE FALSE
## 5 FALSE FALSE FALSE
## 6 FALSE FALSE FALSE
## 7 TRUE FALSE FALSE
## 8 TRUE FALSE FALSE
## 9 TRUE FALSE FALSE
## 10 TRUE FALSE FALSE
## 11 TRUE TRUE TRUE
## 12 TRUE TRUE TRUE
## 13 TRUE TRUE TRUE
## Model.TRANSMISSIONContinuosVariable Model.TRANSMISSIONManual Model.YEAR
## 1 FALSE FALSE FALSE
## 2 FALSE FALSE FALSE
## 3 FALSE FALSE FALSE
## 4 FALSE FALSE FALSE
## 5 FALSE FALSE FALSE
## 6 FALSE FALSE TRUE
## 7 FALSE FALSE TRUE
## 8 TRUE FALSE TRUE
## 9 TRUE TRUE TRUE
## 10 TRUE TRUE TRUE
## 11 TRUE TRUE TRUE
## 12 TRUE TRUE TRUE
## 13 TRUE TRUE TRUE
## Rsq AdRsq CP BIC RSS AIC MSE1
## 1 0.8752 0.8751 441445.4143 -43383.26 10346034.6 -43399.15 496.0224
## 2 0.9874 0.9874 25744.4960 -91239.69 1042847.0 -91263.52 49.9999
## 3 0.9889 0.9889 20160.8681 -93893.20 917843.9 -93924.98 44.0086
## 4 0.9906 0.9906 14023.4985 -97265.88 780448.4 -97305.61 37.4226
## 5 0.9938 0.9938 2278.7587 -105823.91 517563.3 -105871.58 24.8184
## 6 0.9943 0.9943 408.0221 -107575.47 475652.6 -107631.09 22.8098
## 7 0.9943 0.9943 115.2182 -107856.88 469055.0 -107920.45 22.4945
## 8 0.9944 0.9944 41.7123 -107922.22 467365.2 -107993.73 22.4145
## 9 0.9944 0.9944 25.6971 -107930.28 466962.1 -108009.74 22.3963
## 10 0.9944 0.9944 23.9186 -107924.11 466877.5 -108011.51 22.3933
## 11 0.9944 0.9944 12.6831 -107927.40 466581.3 -108022.75 22.3801
## 12 0.9944 0.9944 12.0280 -107920.12 466521.9 -108023.41 22.3784
## 13 0.9944 0.9944 14.0000 -107910.20 466521.3 -108021.44 22.3794Plots
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
Insights
- All the screening methods had the same result:
EMISSIONS ~ CONSUMPTION+FUEL+YEAR+GEARS+TRANSMISSION+ENGINE
- The adjusted r-squared was very high for all the options reviewed in the all possible method, which could lead to considering smaller models.
Model 7: result from screening methods
model7 = lm(data = co2_train,
formula = EMISSIONS ~ CONSUMPTION+FUEL+YEAR+GEARS+TRANSMISSION+ENGINE)
s_model7 = summary(model7)
s_model7##
## Call:
## lm(formula = EMISSIONS ~ CONSUMPTION + FUEL + YEAR + GEARS +
## TRANSMISSION + ENGINE, data = co2_train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -66.248 -1.279 -0.135 1.029 52.924
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.707e+02 1.389e+01 -12.291 < 2e-16 ***
## CONSUMPTION 2.288e+01 2.157e-02 1061.064 < 2e-16 ***
## FUELEtanol -1.511e+02 3.340e-01 -452.445 < 2e-16 ***
## FUELNaturalGas -1.033e+02 9.705e-01 -106.430 < 2e-16 ***
## FUELPremium -3.515e+01 2.706e-01 -129.916 < 2e-16 ***
## FUELRegular -3.463e+01 2.700e-01 -128.237 < 2e-16 ***
## YEAR 1.022e-01 6.961e-03 14.680 < 2e-16 ***
## GEARS 6.622e-01 3.699e-02 17.902 < 2e-16 ***
## TRANSMISSIONAutomatic -6.357e-01 1.811e-01 -3.511 0.000447 ***
## TRANSMISSIONAutomaticSShift -6.980e-01 1.752e-01 -3.984 6.81e-05 ***
## TRANSMISSIONContinuosVariable 1.552e+00 2.977e-01 5.213 1.87e-07 ***
## TRANSMISSIONManual -9.862e-01 1.830e-01 -5.388 7.20e-08 ***
## ENGINE -7.035e-02 4.317e-02 -1.629 0.103227
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.731 on 20847 degrees of freedom
## Multiple R-squared: 0.9944, Adjusted R-squared: 0.9944
## F-statistic: 3.069e+05 on 12 and 20847 DF, p-value: < 2.2e-16Detecting unequal Variance
Insights
- In the residual plot, it can be seen a pattern out of the 2*REE threshold.
Model 8: Model 7 plus Interaction
model8 = lm(data=co2_train,
formula = EMISSIONS ~CONSUMPTION+FUEL+YEAR+GEARS+TRANSMISSION+ENGINE+CONSUMPTION*FUEL)
s_model8 = summary(model8)
s_model8##
## Call:
## lm(formula = EMISSIONS ~ CONSUMPTION + FUEL + YEAR + GEARS +
## TRANSMISSION + ENGINE + CONSUMPTION * FUEL, data = co2_train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -66.969 -1.065 -0.110 0.854 25.564
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.639e+02 5.426e+00 -67.078 < 2e-16 ***
## CONSUMPTION 2.732e+01 3.649e-02 748.607 < 2e-16 ***
## FUELEtanol 9.815e-01 5.042e-01 1.947 0.05158 .
## FUELNaturalGas -6.717e+00 2.246e+00 -2.990 0.00279 **
## FUELPremium 6.546e-01 3.617e-01 1.810 0.07029 .
## FUELRegular 1.506e+00 3.575e-01 4.212 2.54e-05 ***
## YEAR 1.791e-01 2.713e-03 65.994 < 2e-16 ***
## GEARS 3.677e-01 1.437e-02 25.597 < 2e-16 ***
## TRANSMISSIONAutomatic -3.043e-01 7.030e-02 -4.329 1.51e-05 ***
## TRANSMISSIONAutomaticSShift -5.601e-01 6.792e-02 -8.247 < 2e-16 ***
## TRANSMISSIONContinuosVariable 1.329e+00 1.155e-01 11.509 < 2e-16 ***
## TRANSMISSIONManual -4.365e-01 7.097e-02 -6.151 7.82e-10 ***
## ENGINE -5.027e-01 1.682e-02 -29.890 < 2e-16 ***
## CONSUMPTION:FUELEtanol -1.101e+01 4.166e-02 -264.197 < 2e-16 ***
## CONSUMPTION:FUELNaturalGas -7.663e+00 1.354e-01 -56.585 < 2e-16 ***
## CONSUMPTION:FUELPremium -3.917e+00 3.656e-02 -107.147 < 2e-16 ***
## CONSUMPTION:FUELRegular -3.953e+00 3.626e-02 -109.029 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.833 on 20843 degrees of freedom
## Multiple R-squared: 0.9992, Adjusted R-squared: 0.9992
## F-statistic: 1.54e+06 on 16 and 20843 DF, p-value: < 2.2e-16Detecting unequal Variance
Insights
The term with the interaction eliminated the pattern in the residual plot.
The model has a high r-squared, a small standard error, and all the coefficients are significant.
Modeling based on Descriptive Analysis
Model 1: Numerical Variables
model1 = lm(data = co2_train,
formula = EMISSIONS~CONSUMPTION+CYLINDERS+ENGINE)
s_model1 = summary(model1)
s_model1##
## Call:
## lm(formula = EMISSIONS ~ CONSUMPTION + CYLINDERS + ENGINE, data = co2_train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -130.015 -4.710 2.010 9.593 83.500
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.00429 0.66105 51.44 <2e-16 ***
## CONSUMPTION 17.00236 0.08187 207.68 <2e-16 ***
## CYLINDERS 3.33742 0.19997 16.69 <2e-16 ***
## ENGINE 3.31655 0.28966 11.45 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 21.36 on 20856 degrees of freedom
## Multiple R-squared: 0.8851, Adjusted R-squared: 0.8851
## F-statistic: 5.357e+04 on 3 and 20856 DF, p-value: < 2.2e-16Detecting unequal Variance
Model 2: Numerical Variables without Consumption
model2 = lm(data = co2_train,
formula = EMISSIONS~CYLINDERS+ENGINE)
s_model2 = summary(model2)
s_model2##
## Call:
## lm(formula = EMISSIONS ~ CYLINDERS + ENGINE, data = co2_train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -176.120 -24.208 -2.656 22.328 236.559
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 116.1766 0.9275 125.25 <2e-16 ***
## CYLINDERS 8.3211 0.3477 23.93 <2e-16 ***
## ENGINE 27.3415 0.4651 58.78 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 37.42 on 20857 degrees of freedom
## Multiple R-squared: 0.6476, Adjusted R-squared: 0.6475
## F-statistic: 1.916e+04 on 2 and 20857 DF, p-value: < 2.2e-16Detecting unequal Variance
Model 3: Numerical and Categorical
model3 = lm(data = co2_train,
formula=EMISSIONS~CONSUMPTION+CYLINDERS+ENGINE+FUEL+TRANSMISSION)
s_model3 = summary(model3)
s_model3##
## Call:
## lm(formula = EMISSIONS ~ CONSUMPTION + CYLINDERS + ENGINE + FUEL +
## TRANSMISSION, data = co2_train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -64.196 -1.433 -0.630 0.830 52.724
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 41.25178 0.35260 116.992 < 2e-16 ***
## CONSUMPTION 22.70103 0.02190 1036.456 < 2e-16 ***
## CYLINDERS 0.01668 0.04938 0.338 0.73560
## ENGINE 0.21394 0.06898 3.101 0.00193 **
## FUELEtanol -150.43381 0.34118 -440.924 < 2e-16 ***
## FUELNaturalGas -104.65314 1.00839 -103.783 < 2e-16 ***
## FUELPremium -35.28884 0.28136 -125.422 < 2e-16 ***
## FUELRegular -35.40093 0.27815 -127.275 < 2e-16 ***
## TRANSMISSIONAutomatic -2.53600 0.18159 -13.966 < 2e-16 ***
## TRANSMISSIONAutomaticSShift -1.14143 0.18117 -6.300 3.03e-10 ***
## TRANSMISSIONContinuosVariable -2.52036 0.23978 -10.511 < 2e-16 ***
## TRANSMISSIONManual -2.73779 0.18394 -14.884 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.922 on 20848 degrees of freedom
## Multiple R-squared: 0.9939, Adjusted R-squared: 0.9939
## F-statistic: 3.091e+05 on 11 and 20848 DF, p-value: < 2.2e-16Insights
- Cylinders is not significant, because of the high correlation with engine.
Detecting unequal Variance
Model 4: Model3 without Cylinders
model4 = lm(data = co2_train,
formula=EMISSIONS~CONSUMPTION+ENGINE+FUEL+TRANSMISSION)
s_model4 = summary(model4)
s_model4##
## Call:
## lm(formula = EMISSIONS ~ CONSUMPTION + ENGINE + FUEL + TRANSMISSION,
## data = co2_train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -64.218 -1.430 -0.630 0.825 52.728
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 41.27969 0.34278 120.427 < 2e-16 ***
## CONSUMPTION 22.70222 0.02161 1050.401 < 2e-16 ***
## ENGINE 0.23186 0.04408 5.260 1.46e-07 ***
## FUELEtanol -150.44115 0.34048 -441.852 < 2e-16 ***
## FUELNaturalGas -104.66562 1.00769 -103.867 < 2e-16 ***
## FUELPremium -35.28394 0.28098 -125.575 < 2e-16 ***
## FUELRegular -35.40427 0.27796 -127.370 < 2e-16 ***
## TRANSMISSIONAutomatic -2.53977 0.18124 -14.013 < 2e-16 ***
## TRANSMISSIONAutomaticSShift -1.14454 0.18093 -6.326 2.57e-10 ***
## TRANSMISSIONContinuosVariable -2.52406 0.23952 -10.538 < 2e-16 ***
## TRANSMISSIONManual -2.74310 0.18326 -14.968 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.922 on 20849 degrees of freedom
## Multiple R-squared: 0.9939, Adjusted R-squared: 0.9939
## F-statistic: 3.4e+05 on 10 and 20849 DF, p-value: < 2.2e-16Detecting unequal Variance
Model 5: Model4+Interaction
model5 = lm(data = co2_train,
formula=EMISSIONS~CONSUMPTION+ENGINE+FUEL+TRANSMISSION+CONSUMPTION*FUEL)
s_model5 = summary(model5)
s_model5##
## Call:
## lm(formula = EMISSIONS ~ CONSUMPTION + ENGINE + FUEL + TRANSMISSION +
## CONSUMPTION * FUEL, data = co2_train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -66.570 -1.269 -0.704 0.460 26.823
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.39748 0.46032 3.036 0.00240 **
## CONSUMPTION 27.07171 0.04714 574.231 < 2e-16 ***
## ENGINE -0.13952 0.02138 -6.526 6.93e-11 ***
## FUELEtanol 0.41456 0.65268 0.635 0.52533
## FUELNaturalGas -8.97395 2.90822 -3.086 0.00203 **
## FUELPremium 0.11155 0.46815 0.238 0.81166
## FUELRegular 1.35406 0.46234 2.929 0.00341 **
## TRANSMISSIONAutomatic -2.51424 0.08769 -28.671 < 2e-16 ***
## TRANSMISSIONAutomaticSShift -1.21899 0.08730 -13.964 < 2e-16 ***
## TRANSMISSIONContinuosVariable -1.46664 0.11591 -12.654 < 2e-16 ***
## TRANSMISSIONManual -2.59580 0.08845 -29.349 < 2e-16 ***
## CONSUMPTION:FUELEtanol -10.88368 0.05387 -202.022 < 2e-16 ***
## CONSUMPTION:FUELNaturalGas -7.58262 0.17534 -43.246 < 2e-16 ***
## CONSUMPTION:FUELPremium -3.86530 0.04733 -81.668 < 2e-16 ***
## CONSUMPTION:FUELRegular -3.98229 0.04694 -84.833 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.374 on 20845 degrees of freedom
## Multiple R-squared: 0.9986, Adjusted R-squared: 0.9986
## F-statistic: 1.049e+06 on 14 and 20845 DF, p-value: < 2.2e-16INSIGHTS
- Engine turned negative
Detecting unequal Variance
Model 6: Model5-Engine
model6 = lm(data = co2_train,
formula=EMISSIONS~CONSUMPTION+FUEL+TRANSMISSION+CONSUMPTION*FUEL)
s_model6 = summary(model6)
s_model6##
## Call:
## lm(formula = EMISSIONS ~ CONSUMPTION + FUEL + TRANSMISSION +
## CONSUMPTION * FUEL, data = co2_train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -66.442 -1.261 -0.701 0.468 26.938
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.55471 0.46014 3.379 0.00073 ***
## CONSUMPTION 27.01290 0.04632 583.168 < 2e-16 ***
## FUELEtanol 0.39252 0.65332 0.601 0.54797
## FUELNaturalGas -9.07724 2.91108 -3.118 0.00182 **
## FUELPremium 0.15054 0.46858 0.321 0.74800
## FUELRegular 1.33542 0.46279 2.886 0.00391 **
## TRANSMISSIONAutomatic -2.55002 0.08761 -29.107 < 2e-16 ***
## TRANSMISSIONAutomaticSShift -1.25108 0.08724 -14.340 < 2e-16 ***
## TRANSMISSIONContinuosVariable -1.54475 0.11540 -13.386 < 2e-16 ***
## TRANSMISSIONManual -2.59199 0.08853 -29.277 < 2e-16 ***
## CONSUMPTION:FUELEtanol -10.86745 0.05387 -201.734 < 2e-16 ***
## CONSUMPTION:FUELNaturalGas -7.56915 0.17550 -43.130 < 2e-16 ***
## CONSUMPTION:FUELPremium -3.86477 0.04738 -81.576 < 2e-16 ***
## CONSUMPTION:FUELRegular -3.97446 0.04697 -84.609 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.376 on 20846 degrees of freedom
## Multiple R-squared: 0.9986, Adjusted R-squared: 0.9986
## F-statistic: 1.128e+06 on 13 and 20846 DF, p-value: < 2.2e-16Insights
- Some fuel categories became not significant by their own (without interaction).
Detecting unequal Variance
Model 9: Simple Model
model9 = lm(data = co2_train,
formula=EMISSIONS~CONSUMPTION+FUEL+CONSUMPTION*FUEL)
s_model9 = summary(model9)
s_model9##
## Call:
## lm(formula = EMISSIONS ~ CONSUMPTION + FUEL + CONSUMPTION * FUEL,
## data = co2_train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -65.617 -1.418 -1.025 0.987 26.400
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.23092 0.47187 -0.489 0.624586
## CONSUMPTION 26.96563 0.04815 559.982 < 2e-16 ***
## FUELEtanol 0.66427 0.68014 0.977 0.328742
## FUELNaturalGas -9.34275 3.03095 -3.082 0.002056 **
## FUELPremium 0.11727 0.48782 0.240 0.810020
## FUELRegular 1.59207 0.48067 3.312 0.000927 ***
## CONSUMPTION:FUELEtanol -10.86696 0.05608 -193.774 < 2e-16 ***
## CONSUMPTION:FUELNaturalGas -7.54866 0.18272 -41.312 < 2e-16 ***
## CONSUMPTION:FUELPremium -3.82005 0.04930 -77.488 < 2e-16 ***
## CONSUMPTION:FUELRegular -3.98968 0.04885 -81.679 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.474 on 20850 degrees of freedom
## Multiple R-squared: 0.9985, Adjusted R-squared: 0.9985
## F-statistic: 1.502e+06 on 9 and 20850 DF, p-value: < 2.2e-16## The following objects are masked from co2_train (pos = 5):
##
## CONSUMPTION, CYLINDERS, EMISSIONS, ENGINE, FUEL, GEARS,
## TRANSMISSION, YEARDetecting unequal Variance
Conclusion
- It was decided to use the model 9 as the chosen model from the descriptive analysis to prioritize the small amount of predictors, and also, model 9 meets of the assumption and has a high adjusted r-square (0.99).
Testing
Model 8
Model 9
Comparisson between 8 and 9
Conclusions
The model obtained from the variable screening method had a pattern in the residual plot, which was eliminated with the interaction between consumption and type of fuel (Model8). This relationship was discovered in the descriptive analysis, and when applying to the suggested model improved the r-squared from 0.994 to 0.9992.
The chosen model based on the information gathered in the descriptive analysis (Model9) only has three coefficients, consumption, type of fuel and its interactions, but still has an adjusted r squared of 0.9985 which is highly relevant.
As seen in the comparison of the results, when using the testing data, Model 8 and model 9 have similar results which is reasonable as Model 8 has an RMSE of 1.83 and Model 9 has an RMSE of 2.47.
Therefore, the team select to use Model 9 to predict car emissions for its high accuracy, but above all becuase of is simplicity.